A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with M-matrix
Consider the problem of finding the maximal nonpositive solvent $ \varPhi $ of the quadratic matrix equation (QME) $ X^2 + BX + C = 0 $ with $ B $ being a nonsingular $ M $-matrix and $ C $ an $ M $-matrix such that $ B^{-1}C\ge 0 $. Such QME arises from an overdamped vibrating system. Recently, und...
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| Format: | Article |
| Language: | English |
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AIMS Press
2022-02-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2022030?viewType=HTML |
| _version_ | 1798031101854220288 |
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| author | Cairong Chen |
| author_facet | Cairong Chen |
| author_sort | Cairong Chen |
| collection | DOAJ |
| description | Consider the problem of finding the maximal nonpositive solvent $ \varPhi $ of the quadratic matrix equation (QME) $ X^2 + BX + C = 0 $ with $ B $ being a nonsingular $ M $-matrix and $ C $ an $ M $-matrix such that $ B^{-1}C\ge 0 $. Such QME arises from an overdamped vibrating system. Recently, under the condition that $ B - C - I $ is a nonsingular $ M $-matrix, Yu et al. (<italic>Appl. Math. Comput.</italic>, 218 (2011): 3303–3310) proved that $ \rho(\varPhi)\le 1 $ for this QME. In this paper, under the same condition, we slightly improve their result and prove that $ \rho(\varPhi) < 1 $, which is important for the quadratic convergence of the structure-preserving doubling algorithm. Then, a new globally monotonically and quadratically convergent structure-preserving doubling algorithm for solving the QME is developed. Numerical examples are presented to demonstrate the feasibility and effectiveness of our method. |
| first_indexed | 2024-04-11T19:50:52Z |
| format | Article |
| id | doaj.art-e544228c151e4df8b483392c1b787d7f |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-04-11T19:50:52Z |
| publishDate | 2022-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-e544228c151e4df8b483392c1b787d7f2022-12-22T04:06:18ZengAIMS PressElectronic Research Archive2688-15942022-02-0130257458410.3934/era.2022030A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with M-matrixCairong Chen 0School of Mathematics and Statistics, FJKLMAA and Center for Applied Mathematics of Fujian Province, Fujian Normal University, Fuzhou 350007, ChinaConsider the problem of finding the maximal nonpositive solvent $ \varPhi $ of the quadratic matrix equation (QME) $ X^2 + BX + C = 0 $ with $ B $ being a nonsingular $ M $-matrix and $ C $ an $ M $-matrix such that $ B^{-1}C\ge 0 $. Such QME arises from an overdamped vibrating system. Recently, under the condition that $ B - C - I $ is a nonsingular $ M $-matrix, Yu et al. (<italic>Appl. Math. Comput.</italic>, 218 (2011): 3303–3310) proved that $ \rho(\varPhi)\le 1 $ for this QME. In this paper, under the same condition, we slightly improve their result and prove that $ \rho(\varPhi) < 1 $, which is important for the quadratic convergence of the structure-preserving doubling algorithm. Then, a new globally monotonically and quadratically convergent structure-preserving doubling algorithm for solving the QME is developed. Numerical examples are presented to demonstrate the feasibility and effectiveness of our method.https://www.aimspress.com/article/doi/10.3934/era.2022030?viewType=HTMLquadratic matrix equationstructure-preserving doubling algorithmm-matrixmaximal nonpositive solventquadratic convergence |
| spellingShingle | Cairong Chen A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with M-matrix Electronic Research Archive quadratic matrix equation structure-preserving doubling algorithm m-matrix maximal nonpositive solvent quadratic convergence |
| title | A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with M-matrix |
| title_full | A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with M-matrix |
| title_fullStr | A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with M-matrix |
| title_full_unstemmed | A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with M-matrix |
| title_short | A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with M-matrix |
| title_sort | structure preserving doubling algorithm for solving a class of quadratic matrix equation with m matrix |
| topic | quadratic matrix equation structure-preserving doubling algorithm m-matrix maximal nonpositive solvent quadratic convergence |
| url | https://www.aimspress.com/article/doi/10.3934/era.2022030?viewType=HTML |
| work_keys_str_mv | AT cairongchen astructurepreservingdoublingalgorithmforsolvingaclassofquadraticmatrixequationwithmmatrix AT cairongchen structurepreservingdoublingalgorithmforsolvingaclassofquadraticmatrixequationwithmmatrix |